Anomalous angular dependence of the upper critical induction of orthorhombic ferromagnetic superconductors with completely broken p-wave symmetry

Abstract

We employ the Klemm-Clem transformations to map the equations of motion for the Green functions of a clean superconductor with a general ellipsoidal Fermi surface (FS) characterized by the effective masses $m_1,m_2$, and $m_3$ in the presence of an arbitrarily directed magnetic induction $\mathbf{B}=B(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$ onto those of a spherical FS. We then obtain the transformed gap equation for a transformed pairing interaction $\tilde{V}(\widehat{\tilde{\mathbf{k}}},{\widehat{\tilde{\mathbf{k}}}}^{\prime })$ appropriate for any orbital order parameter symmetry. We use these results to calculate the upper critical induction $B_{c2}(\theta ,\varphi )$ for an orthorhombic ferromagnetic superconductor with transition temperatures $T_{\mathrm{Curie}}>T_c$. We assume the FS is split by strong spin-orbit coupling, with a single parallel-spin ($\uparrow \uparrow$) pairing interaction of the \textit{p}-wave polar state form locked onto the ${\hat{\mathbf{e}}}_3$ crystal axis normal to the spontaneous magnetization ${\mathbf{M}}_0\perp {\hat{\mathbf{e}}}_3$ due to the ferromagnetism. The orbital harmonic oscillator eigenvalues are modified according to $B\to B\alpha$, where $\alpha (\theta ,\varphi )=\sqrt{m_3/m}\sqrt{{\cos }^2\theta +{\gamma }^{-2}(\varphi ){\sin }^2\theta }$, ${\gamma }^2(\varphi )=m_3/(m_1{\cos }^2\varphi +m_2{\sin }^2\varphi )$ and $m=(m_1{m}_2{m}_3{)}^{1/3}$. At fixed $\varphi$, the order parameter anisotropy causes $B_{c2}$ to exhibit a novel $\theta$-dependence, which for ${\gamma }^2(\varphi )>3$ becomes a double peak at $0^{\circ }<{\theta }^{\ast }<90^{\circ }$ and at $180^{\circ }-{\theta }^{\ast }$, providing a sensitive bulk test of the order parameter orbital symmetry in both phases of URhGe and in similar compounds still to be discovered.

12 pages 5 figures, accepted for publication in Phys. Rev. B

References in corpus (5)

• Superconductivity on the border of weak itinerant ferromagnetism in UCoGe
• The Order Parameter for the Superconducting Phases of UPt${}_3$
• Bulk superconducting phase with a full energy gap in the doped topological insulator Cu_xBi_2Se_3
• First-Order Superconducting Transition of Sr2RuO4
• Nodeless spin triplet superconducting gap in Sr2RuO4

Cited by in corpus (6)

• Is the anisotropy of the upper critical field of Sr${}_2$RuO${}_4$ consistent with a helical $p$-wave state?
• Angular variation of the magnetoresistance of the superconducting ferromagnet UCoGe
• Ground-state phase diagram of the $S=1$ one-dimensional Kondo lattice model with a uniaxial anisotropy under transverse fields
• Angular dependence of the upper critical induction of clean $s$- and $d_{x^2-y^2}$-wave superconductors with self-consistent ellipsoidal effective mass and Zeeman anisotropies
• Towards a Microscopic Theory of the Knight Shift in an Anisotropic, Multiband Type-II Superconductor
• First-order chiral to non-chiral transition in the angular dependence of the upper critical induction of the Scharnberg-Klemm $p$-wave pair state